FUNCTIONS, LIMITS, AND CONTINUITY in .NET

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FUNCTIONS, LIMITS, AND CONTINUITY
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If m @ f x @ M in an interval, we call f x bounded. Frequencly, when we wish to indicate that a function is bounded, we shall write j f x j < P.
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EXAMPLES. 1. 2. f x 3 x is bounded in 1 @ x @ 1. An upper bound is 4 (or any number greater than 4). A lower bound is 2 (or any number less than 2). f x 1=x is not bounded in 0 < x < 4 since by choosing x su ciently close to zero, f x can be made as large as we wish, so that there is no upper bound. However, a lower bound is given by 1 1 4 (or any number less than 4).
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If f x has an upper bound it has a least upper bound (l.u.b.); if it has a lower bound it has a greatest lower bound (g.l.b.). (See 1 for these de nitions.)
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MONOTONIC FUNCTIONS A function is called monotonic increasing in an interval if for any two points x1 and x2 in the interval such that x1 < x2 , f x1 @ f x2 . If f x1 < f x2 the function is called strictly increasing. Similarly if f x1 A f x2 whenever x1 < x2 , then f x is monotonic decreasing; while if f x1 > f x2 , it is strictly decreasing.
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INVERSE FUNCTIONS.
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Suppose y is the range variable of a function f with domain variable x. Furthermore, let the correspondence between the domain and range values be one-to-one. Then a new function f 1 , called the inverse function of f , can be created by interchanging the domain and range of f . This information is contained in the form x f 1 y . As you work with the inverse function, it often is convenient to rename the domain variable as x and use y to symbolize the images, then the notation is y f 1 x . In particular, this allows graphical expression of the inverse function with its domain on the horizontal axis. Note: f 1 does not mean f to the negative one power. When used with functions the notation f 1 always designates the inverse function to f . If the domain and range elements of f are not in one-to-one correspondence (this would mean that distinct domain elements have the same image), then a collection of one-to-one functions may be created. Each of them is called a branch. It is often convenient to choose one of these branches, called the principal branch, and denote it as the inverse function, f 1 . The range values of f that compose the principal branch, and hence the domain of f 1 , are called the principal values. (As will be seen in the section of elementary functions, it is common practice to specify these principal values for that class of functions.)
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EXAMPLE. Suppose f is generated by y sin x and the domain is 1 @ x @ 1. Then there are an in nite number of domain values that have the same image. (A nite portion of the graph is illustrated below in Fig. 3-2(a.) In Fig. 3-2(b) the graph is rotated about a line at 458 so that the x-axis rotates into the y-axis. Then the variables are interchanged so that the x-axis is once again the horizontal one. We see that the image of an x value is not unique. Therefore, a set of principal values must be chosen to establish an inverse function. A choice of a branch is   accomplished by restricting the domain of the starting function, sin x. For example, choose @ x @ . 2 2 Then there is a one-to-one correspondence between the elements of this domain and the images in 1 @ x @ 1. Thus, f 1 may be de ned with this interval as its domain. This idea is illustrated in Fig. 3-2(c) and Fig. 3-2(d). With the domain of f 1 represented on the horizontal axis and by the variable x, we write y sin 1 x, 1 @ x @ 1.  If x 1, then the corresponding range value is y . 2 6 1 Note: In algebra, b 1 means and the fact that bb 1 produces the identity element 1 is simply a rule of algebra b generalized from arithmetic. Use of a similar exponential notation for inverse functions is justi ed in that corresponding algebraic characteristics are displayed by f 1 f x x and f f 1 x x.
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